Advanced SearchTechniques for Large Scale Data Analytics
Pavel Zezula and Jan Sedmidubsky
Masaryk University
http://disa.fi.muni.cz
 Given a cloud of data points we want to
understand its structure
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 2
3
 Given a set of points, with a notion of distance
between points, group the points into some
number of clusters, so that
▪ Members of a cluster are close/similar to each other
▪ Members of different clusters are dissimilar
 Usually:
▪ Points are in a high-dimensional space
▪ Similarity is defined using a distance measure
▪ Euclidean, Cosine, Jaccard, edit distance, …
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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x x
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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Outlier Cluster
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 5
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 Clustering in two dimensions looks easy
 Clustering small amounts of data looks easy
 And in most cases, looks are not deceiving
 Many applications involve not 2, but 10 or
10,000 dimensions
 High-dimensional spaces look different:
Almost all pairs of points are at about the
same distance
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 Intuitively: Music divides into categories, and
customers prefer a few categories
▪ But what are categories really?
 Represent a CD by a set of customers who
bought it:
 Similar CDs have similar sets of customers,
and vice-versa
7Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
Space of all CDs:
 Think of a space with one dim. for each
customer
▪ Values in a dimension may be 0 or 1 only
▪ A CD is a point in this space (x1, x2,…, xk),
where xi = 1 iff the i th customer bought the CD
 For Amazon, the dimension is tens of millions
 Task: Find clusters of similar CDs
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 8
Finding topics:
 Represent a document by a vector
(x1, x2,…, xk), where xi = 1 iff the i th word
(in some order) appears in the document
▪ It actually doesn’t matter if k is infinite; i.e., we
don’t limit the set of words
 Documents with similar sets of words
may be about the same topic
9Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 As with CDs we have a choice when we
think of documents as sets of words or
shingles:
▪ Sets as vectors: Measure similarity by the
cosine distance
▪ Sets as sets: Measure similarity by the
Jaccard distance
▪ Sets as points: Measure similarity by
Euclidean distance
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 10
11
 Hierarchical:
▪ Agglomerative (bottom up):
▪ Initially, each point is a cluster
▪ Repeatedly combine the two
“nearest” clusters into one
▪ Divisive (top down):
▪ Start with one cluster and recursively split it
 Point assignment:
▪ Maintain a set of clusters
▪ Points belong to “nearest” cluster
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 Key operation:
Repeatedly combine
two nearest clusters
 Three important questions:
▪ 1) How do you represent a cluster of more
than one point?
▪ 2) How do you determine the “nearness” of
clusters?
▪ 3) When to stop combining clusters?
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 12
 Key operation: Repeatedly combine two
nearest clusters
 (1) How to represent a cluster of many points?
▪ Key problem: As you merge clusters, how do you
represent the “location” of each cluster, to tell which
pair of clusters is closest?
 Euclidean case: each cluster has a
centroid = average of its (data)points
 (2) How to determine “nearness” of clusters?
▪ Measure cluster distances by distances of centroids
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 13
14
(5,3)
o
(1,2)
o
o (2,1) o (4,1)
o (0,0) o (5,0)
x (1.5,1.5)
x (4.5,0.5)
x (1,1)
x (4.7,1.3)
Data:
o … data point
x … centroid
Dendrogram
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
What about the Non-Euclidean case?
 The only “locations” we can talk about are the
points themselves
▪ i.e., there is no “average” of two points
 Approach 1:
▪ (1) How to represent a cluster of many points?
clustroid = (data)point “closest” to other points
▪ (2) How do you determine the “nearness” of
clusters? Treat clustroid as if it were centroid, when
computing inter-cluster distances
15Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 (1) How to represent a cluster of many points?
clustroid = point “closest” to other points
 Possible meanings of “closest”:
▪ Smallest maximum distance to other points
▪ Smallest average distance to other points
▪ Smallest sum of squares of distances to other points
▪ For distance metric d clustroid c of cluster C is:
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 16
Cx
c
cxd 2
),(min
Centroid is the avg. of all (data)points
in the cluster. This means centroid is
an “artificial” point.
Clustroid is an existing (data)point
that is “closest” to all other points in
the cluster.
X
Cluster on
3 datapoints
Centroid
Clustroid
Datapoint
 (2) How do you determine the “nearness” of
clusters?
▪ Approach 2:
Intercluster distance = minimum of the distances
between any two points, one from each cluster
▪ Approach 3:
Pick a notion of “cohesion” of clusters, e.g.,
maximum distance from the clustroid
▪ Merge clusters whose union is most cohesive
17Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 Approach 3.1: Use the diameter of the
merged cluster = maximum distance between
points in the cluster
 Approach 3.2: Use the average distance
between points in the cluster
 Approach 3.3: Use a density-based approach
▪ Take the diameter or avg. distance, e.g., and divide
by the number of points in the cluster
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 18
 Naïve implementation of hierarchical
clustering:
▪ At each step, compute pairwise distances
between all pairs of clusters, then merge
▪ O(N3)
 Careful implementation using priority queue
can reduce time to O(N2 log N)
▪ Still too expensive for really big datasets
that do not fit in memory
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 19
 Assumes Euclidean space/distance
 Start by picking k, the number of clusters
 Initialize clusters by picking one point per
cluster
▪ Example: Pick one point at random, then k-1
other points, each as far away as possible from
the previous points
21Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 1) For each point, place it in the cluster whose
current centroid it is nearest
 2) After all points are assigned, update the
locations of centroids of the k clusters
 3) Reassign all points to their closest centroid
▪ Sometimes moves points between clusters
 Repeat 2 and 3 until convergence
▪ Convergence: Points don’t move between clusters
and centroids stabilize
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 22
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 23
x
x
x
x
x
x
x x
x … data point
… centroid
x
x
x
Clusters after round 1
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 24
x
x
x
x
x
x
x x
x … data point
… centroid
x
x
x
Clusters after round 2
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 25
x
x
x
x
x
x
x x
x … data point
… centroid
x
x
x
Clusters at the end
How to select k?
 Try different k, looking at the change in the
average distance to centroid as k increases
 Average falls rapidly until right k, then
changes little
26
k
Average
distance to
centroid
Best value
of k
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 27
x x
x x x x
x x x x
x x x
x x
x
xx x
x x
x x x
x
x x x
x
x x
x x x x
x x x
x
x
x
Too few;
many long
distances
to centroid.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 28
x x
x x x x
x x x x
x x x
x x
x
xx x
x x
x x x
x
x x x
x
x x
x x x x
x x x
x
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x
Just right;
distances
rather short.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 29
x x
x x x x
x x x x
x x x
x x
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xx x
x x
x x x
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x
Too many;
little improvement
in average
distance.
Extension of k-means to large data
 BFR [Bradley-Fayyad-Reina] is a
variant of k-means designed to
handle very large (disk-resident) data sets
 Assumes that clusters are normally distributed
around a centroid in a Euclidean space
▪ Standard deviations in different
dimensions may vary
▪ Clusters are axis-aligned ellipses
 Efficient way to summarize clusters
(want memory required O(clusters) and not O(data))
31Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 Points are read from disk one main-memoryfull
at a time
 Most points from previous memory loads are
summarized by simple statistics
 To begin, from the initial load we select the
initial k centroids by some sensible approach:
▪ Take k random points
▪ Take a small random sample and cluster optimally
▪ Take a sample; pick a random point, and then
k–1 more points, each as far from the previously
selected points as possible
32Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
3 sets of points which we keep track of:
 Discard set (DS):
▪ Points close enough to a centroid to be
summarized
 Compression set (CS):
▪ Groups of points that are close together but
not close to any existing centroid
▪ These points are summarized, but not
assigned to a cluster
 Retained set (RS):
▪ Isolated points waiting to be assigned to a
compression set
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 33
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 34
A cluster. Its points
are in the DS.
The centroid
Compressed sets.
Their points are in
the CS.
Points in
the RS
Discard set (DS): Close enough to a centroid to be summarized
Compression set (CS): Summarized, but not assigned to a cluster
Retained set (RS): Isolated points
For each cluster, the discard set (DS) is
summarized by:
 The number of points, N
 The vector SUM, whose ith component is the
sum of the coordinates of the points in the
ith dimension
 The vector SUMSQ: ith component = sum of
squares of coordinates in ith dimension
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 35
A cluster.
All its points are in the DS. The centroid
 2d + 1 values represent any size cluster
▪ d = number of dimensions
 Average in each dimension (the centroid)
can be calculated as SUMi / N
▪ SUMi = ith component of SUM
 Variance of a cluster’s discard set in
dimension i is: (SUMSQi / N) – (SUMi / N)2
▪ And standard deviation is the square root of that
 Next step: Actual clustering
36Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
Note: Dropping the “axis-aligned” clusters assumption would require
storing full covariance matrix to summarize the cluster. So, instead of
SUMSQ being a d-dim vector, it would be a d x d matrix, which is too big!
Processing the “Memory-Load” of points (1):
 1) Find those points that are “sufficiently
close” to a cluster centroid and add those
points to that cluster and the DS
▪ These points are so close to the centroid that
they can be summarized and then discarded
 2) Use any main-memory clustering algorithm
to cluster the remaining points and the old RS
▪ Clusters go to the CS; outlying points to the RS
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 37
Discard set (DS): Close enough to a centroid to be summarized.
Compression set (CS): Summarized, but not assigned to a cluster
Retained set (RS): Isolated points
Processing the “Memory-Load” of points (2):
 3) DS set: Adjust statistics of the clusters to
account for the new points
▪ Add Ns, SUMs, SUMSQs
 4) Consider merging compressed sets in the CS
 5) If this is the last round, add all compressed
sets in the CS and all RS points into their nearest
cluster
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 38
Discard set (DS): Close enough to a centroid to be summarized.
Compression set (CS): Summarized, but not assigned to a cluster
Retained set (RS): Isolated points
 Q1) How do we decide if a point is “close
enough” to a cluster that we will add the
point to that cluster?
 Q2) How do we decide whether two
compressed sets (CS) deserve to be
combined into one?
39Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
 Q1) We need a way to decide whether to put a new
point into a cluster (and discard)
▪ Using the Mahalanobis distance – accept a point for a
cluster if its M.D. is < some threshold, e.g. 2 𝑑
▪ If clusters are normally distributed in d dimensions, then after
transformation, one standard deviation = 𝒅, i.e., 68% of the points
of the cluster will have a Mahalanobis distance < 𝒅
▪ For point (x1, …, xd) and centroid (c1, …, cd)
1. Normalize in each dimension: yi = (xi - ci) / i
2. Take sum of the squares of the yi
3. Take the square root
𝑑 𝑥, 𝑐 = ෍
𝑖=1
𝑑
𝑦𝑖
2
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 40
σi … standard
deviation of points in
the cluster in the ith
dimension
Q2) Should 2 CS subclusters be combined?
 Compute the variance of the combined
subcluster
▪ N, SUM, and SUMSQ allow us to make that
calculation quickly
 Combine if the combined variance is
below some threshold
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 41
Extension of k-means to clusters
of arbitrary shapes
 Problem with BFR/k-means:
▪ Assumes clusters are normally
distributed in each dimension
▪ And axes are fixed – ellipses at
an angle are not OK
 CURE (Clustering Using REpresentatives):
▪ Assumes a Euclidean distance
▪ Allows clusters to assume any shape
▪ Uses a collection of representative
points to represent clusters
43
Vs.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
2 Pass algorithm. Pass 1:
 0) Pick a random sample of points that fit in
main memory
 1) Initial clusters:
▪ Cluster these points hierarchically – group
nearest points/clusters
 2) Pick representative points:
▪ For each cluster, pick a sample of points, as
dispersed as possible
▪ From the sample, pick representatives by moving
them (say) 20% toward the centroid of the cluster
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 45
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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Pick (say) 4
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cluster.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
Pass 2:
 Now, rescan the whole dataset and
visit each point p in the data set
 Place it in the “closest cluster”
▪ Normal definition of “closest”:
Find the closest representative to p and
assign it to representative’s cluster
49Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
p
 Clustering: Given a set of points, with a notion
of distance between points, group the points
into some number of clusters
 Algorithms:
▪ Agglomerative hierarchical clustering:
▪ Centroid and clustroid
▪ k-means:
▪ Initialization, picking k
▪ BFR
▪ CURE
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 50